Quick and fast are not quite the same. Quick implies a brief burst of speed. In other words, acceleration. Fast implies more sustained speed. In other words, velocity. Quick similes include "quick as a flash" "quick as a wink". Fast similes include "Fast as a jack rabbit in front of a prairie fire" "Fast as a dog will lick a dish". See Bartleby.com. Horses may be considered fast, but they're not generally considered quick, with the possible exception of quarter horses.
This is quite interesting. As a native English speaker, I've never noticed the difference between quick and fast. However, after reading this in its entirety, I have to agree that there are places where only one or the other "feels" right; i.e. "quick" and "fast" are synonyms but have semantic information that differs.